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SageMath
E = EllipticCurve("gt1")
E.isogeny_class()
Elliptic curves in class 43200.gt
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
43200.gt1 | 43200dc3 | \([0, 0, 0, -197100, -44658000]\) | \(-1167051/512\) | \(-371504185344000000\) | \([]\) | \(373248\) | \(2.0785\) | |
43200.gt2 | 43200dc1 | \([0, 0, 0, -5100, 142000]\) | \(-132651/2\) | \(-221184000000\) | \([]\) | \(41472\) | \(0.97993\) | \(\Gamma_0(N)\)-optimal |
43200.gt3 | 43200dc2 | \([0, 0, 0, 18900, 702000]\) | \(9261/8\) | \(-644972544000000\) | \([]\) | \(124416\) | \(1.5292\) |
Rank
sage: E.rank()
The elliptic curves in class 43200.gt have rank \(0\).
Complex multiplication
The elliptic curves in class 43200.gt do not have complex multiplication.Modular form 43200.2.a.gt
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.